The simplest algorithm to solve this regularized UOT problem applies an alternate coordinate ascent on the dual problem, alternately maximizing on \(f\)  and on \(g\) .
We refer to this class of methods as Sinkhorn's algorithm, which was historically derived in the balanced OT case [1]}.
Let us underline that this algorithm does not converge when \(\epsilon =0\)  (because of the constraint in the dual), however it does converge for the regularized case \(\epsilon >0\) .
This algorithm appeared in a variety of fields,
and we refer to [2]} for a historical account.
